Denseness of zero entropy aperiodic ergodic measures

TL;DR

This paper demonstrates that in certain dynamical systems, zero-entropy measures with specific Lyapunov exponents are densely approximated by low-complexity measures, even when periodic measures are absent.

Contribution

It extends the classical density results of periodic measures to nonhyperbolic homoclinic classes lacking the specification property.

Findings

Zero-entropy measures are dense in level sets of ergodic measures with fixed central Lyapunov exponents.

The density of low-complexity measures holds for interior points of the measure spectrum.

Similar results are established for blender-minimal diffeomorphisms within robustly transitive systems.

Abstract

We study partially hyperbolic homoclinic classes of $C^1$-generic diffeomorphisms with a one-dimensional central bundle, so that the central Lyapunov exponent $\chi^c(\mu)$ is well defined for any ergodic measure $\mu$ supported on the class. We focus on nonhyperbolic homoclinic classes supporting ergodic measures with positive, zero, and negative central exponents. For each $\alpha$ and a nontrivial homoclinic class $H$ of a $C^1$-generic diffeomorphism $f$, we consider the level set of measures \[ \mathcal{M}^\alpha_{\mathrm{erg}}(f,H)= \left\{\text{$\mu$ ergodic, supported on $H$, with } \chi^c(\mu)=\alpha \right\}. \] In this generic setting, the range of $\alpha$ for which $\mathcal{M}^\alpha_{\mathrm{erg}}(f,H)$ is nonempty forms a nontrivial closed interval $I$. Since the set of periodic measures is countable, most of these sets contain no periodic measures. We show that for every $\alpha$ in the interior of $I$, the so-called Axiomatized GIKN measures, a class of low-complexity, zero-entropy measures, are dense in $\mathcal{M}^\alpha_{\mathrm{erg}}(f,H)$. This result can be viewed as an analogue of Sigmund's classical density of periodic measures for systems with the specification property, obtained here in a setting where the specification property does not hold and periodic measures are typically absent (in the considered level sets). We also present a similar result for the open class of blender-minimal diffeomorphisms, contained in the class of $C^1$-robustly transitive ones.